Classification of integrable discrete equations of octahedron type

Vsevolod E. Adler, Alexander I. Bobenko, Yuri B. Suris

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


We use the consistency approach to classify discrete integrable three-dimensional equations of the octahedron type. They are naturally treated on the root lattice Q(A 3) and are consistent on the multi-dimensional lattice Q(A N). Our list includes the most prominent representatives of this class, the discrete KP equation and its Schwarzian (multi-ratio) version, as well as three further equations. The combinatorics and geometry of the octahedron-type equations are explained. In particular, the consistency on the four-dimensional Delaunay cells has its origin in the classical Desargues theorem of projective geometry. The main technical tool used for the classification is the so-called tripodal form of the octahedron-type equations.

Original languageEnglish
Pages (from-to)1822-1889
Number of pages68
JournalInternational Mathematics Research Notices
Issue number8
StatePublished - 2012
Externally publishedYes


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